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# lecture03 - Lecture 3 1.3 Row-Echelon Forms 1.4 Gaussian...

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Chapter 1 Linear Systems & Gaussian Elimination 1 Lecture 3 1.3 Row-Echelon Forms 1.4 Gaussian Elimination

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Lecture 3 Announcement 2 Announcement Tutorial/lab balloting (end Wed Jan 20) Tutorial class begins next week (week 3) Tutorial sets are available in IVLE workbin Lab session 1 in week 4 Lab worksheets will be available Practice sessions begin this week SL1 lecture group : Tuesday SL2 lecture group : Thursday Typo list for textbook in Workbin
Chapter 1 Linear Systems & Gaussian Elimination 3 Section 1.3 Row-Echelon Forms Objective • How to identify a row-echelon form (REF) and a reduced row-echelon form (RREF) ? • How to use REF / RREF to get solutions of linear system? • How to tell the number of solutions from REF? Other terminologies leading entries , pivot columns

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Chapter 1 Linear Systems & Gaussian Elimination 4 Definition 1.3.1 1. If there are any rows that consist entirely of zeros , then they are grouped together at the bottom of the matrix . 0 0 0 0 0 0 " " " " " " # # # " " nonzero rows zero rows (if any) An augmented matrix is said to be in row-echelon form if it has the following 2 properties: 0 0 0 0 0 3 1 0 0 0 1 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 1 0 2 1 0 property 1 not satisfied property 1 satisfied How to identify REF?
Chapter 1 Linear Systems & Gaussian Elimination 5 Definition 1.3.1 2. In any two successive non-zero rows, the first nonzero number in the lower row occurs farther to the right than the first nonzero number in the higher row . 0 0 0 0 " " " " two successive rows leading entries 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 1 0 2 1 0 property 2 not satisfied property 2 satisfied 0 0 0 0 0 0 0 0 0 0 1 0 2 1 0 3 1 0 0 0 How to identify REF?

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Chapter 1 Linear Systems & Gaussian Elimination 6 Definition 1.3.1 0 0 0 0 " " % nonzero rows zero rows (if any) leading entries Combining properties 1 and 2: This is a row-echelon form (REF) everything below the “staircase” is 0 How to identify REF?
Chapter 1 Linear Systems & Gaussian Elimination 7 Definition 1.3.1 0 0 0 0 " " % nonzero rows zero rows (if any) leading entries also called pivot point columns that contain pivot points called pivot columns refers to leading entry of a row , not column How to identify REF?

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Chapter 1 Linear Systems & Gaussian Elimination 8 Definition 1.3.1 3. The leading entry of every nonzero row is 1 . An augmented matrix is said to be in reduced row-echelon form ( RREF ) if it is in row-echelon form and has the following properties: 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 1 0 2 1 0 property 3 not satisfied property 3 satisfied 0 0 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 2 1 0
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lecture03 - Lecture 3 1.3 Row-Echelon Forms 1.4 Gaussian...

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